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The Quadratic Character of \(2\)

The number \(2\) is a quadratic residue mod \(p\) if \(p = 8k + 1\) or \(p = 8k + 7\), but not if \(p = 8k + 3\) or \(p = 8k + 5\). This is proved by a simple counting argument, assuming the existence of a primitive root mod \(p\).